Week 7 - Prove by disprove

    During the week, the lectures on proof techniques are being summed up. Through out the few weeks I am starting to get used to the structures and techniques in proofs. They do not look as horrifying as they were few weeks ago;  I start to feel comfortable doing proofs. However, some times there is still one or two specific things that boggles my thinking. 

Up to this point, many of the proofs are well expressed and specified to be proven. However, being introduced this week, I need to start verify the claim first before I can get my hands working. 

I was introduced to disproving a claim this week. It is used when the claim is proven to be false. To work on a disprove, one need to prove the negation of the claim to be True. Since the negation of the claim is true, the claim must be false or it will no be justify in logic. Disproving works in similar structure of a proof. Therefore it was not too hard to get the hang of the work. However, verifying the proofs was the hard part since it takes certain situation to falsify the claim. Too often, its quite consuming to figure out the light bulb. During a in lecture exercise,  I was given a proof and told to fill the proof as much as possible in a short period of time. However, very few ideas popped up. It can also be quite struggling to disprove. As a result, I try to list out many of the facts that the claim provides and find the ones that did not make too much sense.

*That tutorial was a pain.

*The Promised Problem Solving - Folding.

This is one of the problems in the problem solving session. 

The Folding Problem

First, understanding the problem:

The problem is quite simple to comprehend. Take a strip of paper and fold it towards the right direction over and over again. It will result in creases that points up and down. Each group needs to predict the sequence that the creases are presented. 

Information given:

We conclude the input is the number of folds.

The output will be the number of cease that can be either up or down. 

*The information given is quite limited. 

It was announced that there are many ways to solve the problem. However, we only thought of one plan.

Devise a plan: 

Our initial plan is to record the data of the initial five folds to see if we can figure out some pattern. By using the pattern we can then predict the following sequence of the folds. 

Carrying out the plan:

Here are the data that was collected:

Number of Folds	The Pattern of Creases
1	D
2	U        D        D
3	U   U    D  D  U   D  D
4	UUDUUDDDUUDDUDD

As seen from the first three data, there is a symmetric pattern that goes from the middle of the pattern which is a crease point downwards. The symmetric pattern then extends, to the digit beside the middle point. In the following fold that creases will be a mid point as in between a crease upward and downward. In the fourth folds, we confirmed the results and correctly predicted the next few folds. 

Looking Back:

The Questions could have been solved in an other way and have been able to be solved in a more efficient way. There could be an implementation of algorithm that can automatically give you the result needed instead of human integration that is time consuming. 

Week 6 -- The Slack...

     Tests back. I expected myself to jump off the CN tower. However, the score on the test was quite satisfying. Thinking back on the questions on the test they were not as bad as they seem when I first encountered them during the exam. Therefore, it implies that if I do enough practice, any questions on the exam will be easy.
* TBH I was too young too simple to conclude such statement. 

     Fortunately, the Thanksgiving day-off strip three proof-heavy classes into two. Cheers for the turkey. However, through the two classes, many proofs were gone through. It has been tough for me due to the lack of experiences in proofs; I mostly considered math as computations before entering this course. Thus, I get stuck on almost every single proof. Despite the help from the proof structure, I get barricaded by the “thinking part”, to argue with actually proofs and evidences. Therefore, to increase the efficiency in doing proofs I started a new approach. Before I start doing proofs, I read over sample solution of other proofs to flourish myself with the different tricks and proofs. I will spend a average of twenty minutes on a proof before check for answer. As a result, I am not just copying the answer while trying to push myself out of the boundaries of limited thinking. I am satisfied as by the end of the week, I am starting to get the hang of proofs and it is becoming easier to come up with a proof. In addition, I get the direction in handling the proof as I read through the statements, instead of struggling to find a place to start. 

Not so slack week after all.  

Week Five of Death

    The test was not too bad after all, it gave me confident and morale boost. However, I know the buff is not going maintain for a long period of time; there is always a time frame. Due to the test, not too much materials were introduced this week. It cover mainly the structure of proof; the outline is strict and fundamental. However, by doing proofs in the structure insisted allow me to understand the proof more logically as I travel into the different realm of world. In addition, it is suitable for many proofs. Moreover, it amazes me how a simple proof can be completed by the depiction of an example.

For example to prove for existence:    

S : There exists a dog in the zoo such that it is blue. 

In order to prove the statement. The presences of a blue dog is sufficient. 

Structures of proof remain constant for all types of proofs: existence, case, contradiction, etc. 

Being slaughtered by the test, give me a chance to rest and I promise a problem solving in return. 

Week 4 - The Prelude to Execution.

    A third of the semester has gone by and it is about time for the first term test. This week was for us to practise for the test by grinding through the dreadful assignment. To be honest, the assignment questions were not too much stress except a few particular. Along with my group, we were able to knock through most of the questions. However, we got stuck upon the venn diagrams drawing of question four. It was difficult to imagine the diagrams through our minds. Therefore, we constructed a truth table and drew accordingly to table. However, the quantifiers further confused us as difference orders of the quantifiers affect the meaning of statement. Simply by switching the order of the quantifiers can alter the truth of the statement.

For example:  Let N and M = {1, 2, 3, 4,}


For all of the x in N,  there exists a y in M such that x + y = 5
    This statement will always be true.


However...
If we switch the order of quantifier...


There exists a y in M, such that for all x in N, x + y = 5
    This statement will return a false since there is a counter example which depict that x + y != 5

Nevertheless, there are double quantifiers which allows for unsorted order of quantifiers; the order of the quantifiers will not result in the change of the truth.

To sum up the week, we were introduced the basic structure of the proof. The structure was hard to followed in the first place since I used to do math questions in a rather not organized fashion. However, through some practise, I was able to grasp the fundamentals of proof structure and is able to twist myself to think more logically, constantly providing evidence and reasoning in my operations.